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v o [ 1 ] = α size 12{v rSub { size 8{o} } \[ 1 \] =α} {} ,

v o [ 2 ] = ( 1 α ) α + α size 12{v rSub { size 8{o} } \[ 2 \] = \( 1 - α \) α+α} {} ,

v o [ 3 ] = ( 1 α ) 2 α + ( 1 α ) α + α size 12{v rSub { size 8{o} } \[ 3 \] = \( 1 - α \) rSup { size 8{2} } α+ \( 1 - α \) α+α} {} ,

v o [ k ] = α k = 0 n 1 ( 1 α ) k = 1 ( 1 α ) n size 12{v rSub { size 8{o} } \[ k \] =α Sum cSub { size 8{k=0} } cSup { size 8{n - 1} } { \( 1 - α \) rSup { size 8{k} } } =1 - \( 1 - α \) rSup { size 8{n} } } {}

[We have made use of an important formula for the summation of a geometric series which we will prove shortly.] Hence, the solution is

v o [ n ] = ( 1 ( 1 α ) n ) u [ n ] size 12{v rSub { size 8{o} } \[ n \] = \( 1 - \( 1 - α \) rSup { size 8{n} } \) u \[ n \]} {}

The solution

v o [ n ] = ( 1 ( 1 α ) n ) u [ n ] size 12{v rSub { size 8{o} } \[ n \] = \( 1 - \( 1 - α \) rSup { size 8{n} } \) u \[ n \]} {}

is shown plotted below for α size 12{α} {} = 0.25.

4/ Important side issue — the summation of finite geometric series

Suppose

S = n = l k a n = a l + a l + 1 + . . . + a k size 12{S= Sum cSub { size 8{n=l} } cSup { size 8{k} } {a rSup { size 8{n} } } =a rSup { size 8{l} } +a rSup { size 8{l+1} } + "." "." "." +a rSup { size 8{k} } } {}

Then

αS = α n = l k a n = a l + 1 + a l + 2 + . . . + a k + 1 size 12{αS=α Sum cSub { size 8{n=l} } cSup { size 8{k} } {a rSup { size 8{n} } } =a rSup { size 8{l+1} } +a rSup { size 8{l+2} } + "." "." "." +a rSup { size 8{k+1} } } {}

Hence,

( 1 α ) S = ( 1 α ) n = l k a n = a l a k + 1 size 12{ \( 1 - α \) S= \( 1 - α \) Sum cSub { size 8{n=l} } cSup { size 8{k} } {a rSup { size 8{n} } } =a rSup { size 8{l} } - a rSup { size 8{k+1} } } {}

and

S = n = l k a n = a l a k + 1 ( 1 α ) size 12{S= Sum cSub { size 8{n=l} } cSup { size 8{k} } {a rSup { size 8{n} } } = { {a rSup { size 8{l} } - a rSup { size 8{k+1} } } over { \( 1 - α \) } } } {}

Conclusion

  • Difference equations arise in a variety of interesting contexts.
  • Simple difference equations can be solved iteratively.
  • We will develop more systematic and efficient methods to solve arbitrary difference equations.

X. GENERAL LINEAR DIFFERENCE EQUATION

For a DT system, such as the ones shown previously, we can write the relation between an input variable x[n] and an output variable y[n]by a general difference equation of the form

k = 0 K a k y [ n + k ] = l = 0 L b l x [ n + 1 ] size 12{ Sum cSub { size 8{k=0} } cSup { size 8{K} } {a rSub { size 8{k} } y \[ n+k \] } = Sum cSub { size 8{l=0} } cSup { size 8{L} } {b rSub { size 8{l} } x \[ n+1 \]} } {}

We seek a solution of this system for an input that is of the form

x [ n ] = Xz n u [ n ] size 12{x \[ n \] = ital "Xz" rSup { size 8{n} } u \[ n \]} {}

where X and z are in general complex quantities and u[n] is the unit step function. We will also assume that the system is at rest for n<0, so that y[n] = 0 for n<0.

We will seek the solution for n>0 by finding the homogeneous (unforced solution) and then a particular solution (forced solution).

1/ Homogeneous solution

a/ Geometric (exponential) solution

Let the homogeneous solution be y h [ n ] size 12{y rSub { size 8{h} } \[ n \] } {} , then the homogeneous equation is

k = 0 K a k y [ n + k ] = 0 size 12{ Sum cSub { size 8{k=0} } cSup { size 8{K} } {a rSub { size 8{k} } y \[ n+k} \] =0} {}

To solve this equation we assume a solution of the form

y h [ n ] = n size 12{y rSub { size 8{h} } \[ n \] =Aλ rSup { size 8{n} } } {}

Since

y h [ n + k ] = n + k size 12{y rSub { size 8{h} } \[ n+k \] =Aλ rSup { size 8{n+k} } } {}

we have

k = 0 K a k n + k = 0 size 12{ Sum cSub { size 8{k=0} } cSup { size 8{K} } {a rSub { size 8{k} } Aλ rSup { size 8{n+k} } } =0} {}

b/ Characteristic polynomial

The equation can be factored to yield

( k = 0 K a k λ k ) n = 0 size 12{ \( Sum cSub { size 8{k=0} } cSup { size 8{K} } {a rSub { size 8{k} } λ rSup { size 8{k} } \) } Aλ rSup { size 8{n} } =0} {}

We are not interested in the trivial solution

y h [ n ] = n = 0 size 12{y rSub { size 8{h} } \[ n \] =Aλ rSup { size 8{n} } =0} {}

Therefore, we can divide by n size 12{Aλ rSup { size 8{n} } } {} to obtain the characteristic polynomial

k = 0 K a k λ k = 0 size 12{ Sum cSub { size 8{k=0} } cSup { size 8{K} } {a rSub { size 8{k} } λ rSup { size 8{k} } } =0} {}

This polynomial of order K has K roots which can be exposed by writing the polynomial in factored form

k = 1 K ( λ λ k ) = 0 size 12{ Prod cSub { size 8{k=1} } cSup { size 8{K} } { \( λ - λ rSub { size 8{k} } \) } =0} {}

c/ Natural frequencies

The roots of the characteristic polynomial { λ 1 , λ 2 , . . . λ K } size 12{ lbrace λ rSub { size 8{1} } ,λ rSub { size 8{2} } , "." "." "." λ rSub { size 8{K} } rbrace } {} are called natural frequencies. These are frequencies for which there is an output in the absence of an input.

If the natural frequencies are distinct, i.e., if λ i λ k size 12{λ rSub { size 8{i} }<>λ rSub { size 8{k} } } {} for i k size 12{i<>k} {} the most general homogeneous solution has the form

y h [ n ] = k = 1 K A k λ k n size 12{y rSub { size 8{h} } \[ n \] = Sum cSub { size 8{k=1} } cSup { size 8{K} } {A rSub { size 8{k} } λ rSub { size 8{k} rSup { size 8{n} } } } } {}

Two-minute miniquiz problem

Problem 10-1 — Natural frequencies of the ladder network

The electric ladder network shown below has the following difference equation

v o [ n + 1 ] + ( 2 + rg ) v o [ n ] v o [ n 1 ] = rgv i [ n ] size 12{ - v rSub { size 8{o} } \[ n+1 \] + \( 2+ ital "rg" \) v rSub { size 8{o} } \[ n \]- v rSub { size 8{o} } \[ n - 1 \] = ital "rgv" rSub { size 8{i} } \[ n \]} {}

Find the natural frequencies.

Solution

The natural frequencies are determined by the homogeneous solution

v o [ n + 1 ] + ( 2 + rg ) v o [ n ] v o [ n 1 ] = 0 size 12{ - v rSub { size 8{o} } \[ n+1 \] + \( 2+ ital "rg" \) v rSub { size 8{o} } \[ n \]- v rSub { size 8{o} } \[ n - 1 \] =0} {}

Substituting a solution of the form v 0 [ n ] = n size 12{v rSub { size 8{0} } \[ n \] =Aλ rSup { size 8{n} } } {} results in

{} n + 1 + ( 2 + rg ) n n 1 1 = 0 size 12{ - Aλ rSup { size 8{n+1} } + \( 2+ ital "rg" \) Aλ rSup { size 8{n} } - Aλ rSup { size 8{n - 1} } - 1=0} {}

which yields the characteristic polynomial

λ 2 ( 2 + rg ) λ + 1 = 0 size 12{λ rSup { size 8{2} } - \( 2+ ital "rg" \) λ+1=0} {}

whose roots are the characteristic frequencies

λ 1,2 = ( 1 + rg 2 ) ± ( 1 + rg 2 ) 2 1 size 12{λ rSub { size 8{1,2} } = \( 1+ { { ital "rg"} over {2} } \) +- sqrt { \( 1+ { { ital "rg"} over {2} } \) rSup { size 8{2} } - 1} } {}

d/ Natural frequencies of the ladder network

The electric ladder network shown below has the following difference equation

v o [ n + 1 ] + ( 2 + rg ) v o [ n ] v o [ n 1 ] = rgv i [ n ] size 12{ - v rSub { size 8{o} } \[ n+1 \] + \( 2+ ital "rg" \) v rSub { size 8{o} } \[ n \]- v rSub { size 8{o} } \[ n - 1 \] = ital "rgv" rSub { size 8{i} } \[ n \]} {}

The natural frequencies are:

λ 1,2 = ( 1 + rg 2 ) ± ( 1 + rg 2 ) 2 1 size 12{λ rSub { size 8{1,2} } = \( 1+ { { ital "rg"} over {2} } \) +- sqrt { \( 1+ { { ital "rg"} over {2} } \) rSup { size 8{2} } - 1} } {}

The natural frequencies depend on rg,

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Source:  OpenStax, Signals and systems. OpenStax CNX. Jul 29, 2009 Download for free at http://cnx.org/content/col10803/1.1
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